Theoretical Models of Neural Development

Abstract

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

Keywords: Biological Sciences; Developmental Neuroscience; In Silico Biology; Neuroscience; Theoretical Neuroscience.

Graphical abstract

Figure 1

Models of Neurulation and Cortical Folding (A) Experimental data (left column) and simulation results (right column) from stages 14 (top row) and 17 (bottom row) of axolotl neurulation (adapted with permission from Chen and Brodland (2008)). For the right column, yellow represents neural plate tissue, green represents non-neural ectoderm, and blue represents the neural ridges. The height of each image represents approximately 2 mm. (B) A simulation of cortical folding in humans where the brain is treated as a soft elastic solid and the cortex expands tangentially. GW, gestational weeks. Adapted with permission from Tallinen et al. (2016).

Figure 2

Regional Specification and the Role of Noise (A) Layout of key functional areas in the mouse cortex (O'Leary et al., 2007). V1, primary visual cortex; A1, primary auditory cortex; S1, primary somatosensory cortex; F/M, frontal/motor cortex. (B) Some of the molecular gradients that play a role in specifying areal identity in mouse cortex during development. (C) One of the networks of interactions between signaling molecules in the model of Giacomantonio and Goodhill (2010) that generated arealization patterns most consistent with the experimental data. Shaded and unshaded labels indicate genes and proteins, respectively. (D) Schematic of three sources of noise that constrain concentration measurement by a cell. (E) Specification of boundaries in the developing zebrafish hindbrain by a gradient of retinoic acid. Top: schematic representation of the retinoid acid gradient. Any such gradient can only provide noisy information. Bottom: experimental image showing that by 12 hr postfertilization this gradient has nevertheless helped to specify sharply defined borders of the expression of krox20 (green) and hoxb1a (red). Adapted from Zhang et al. (2012).

Figure 3

Neural Migration and Polarization (A) In the developing cortex neurons migrate from their place of birth along radial glia, forming the layers of the cortex in an inside-out fashion. IZ, intermediate zone; VZ, ventricular zone; SVZ, subventricular zone. Time axis is for mouse (E10; embryonic day 10). (B) Left: the results of a single simulation of cortical layer formation in the model of Caffrey et al. (2014). Right: average of 50 simulations, where the vertical axis is distance and horizontal axis is agent density. Reproduced from Caffrey et al. (2014). (C) In general, several neurites sprout from a neuron, and then the longest becomes the axon, whereas the others become dendrites. (D) The mechanism of polarization modeled in Toriyama et al. (2010): accumulation of shootin1 in the growth cone provides a positive feedback loop.

Figure 4

Axon and Dendrite Growth and Guidance (A) Left: Reconstruction of a typical rat cortical pyramidal neuron. Right: A neuron generated with NETMORPH using appropriately tuned model parameters (reproduced with permission from Koene et al. (2009)). (B) Example of simulated growth of axons (black) and dendrites (red) across three cortical layers (reproduced from Zubler and Douglas (2009)). (C) Some key structures and molecules involved in neurite growth and guidance. (D) Number of growth cone point contacts as a function of time from the model of Padmanabhan and Goodhill (2018), corresponding to stochastic oscillations between growth and paused states. (E) Model predictions for whether axons are attracted or repelled in response to a molecular gradient as a function of levels of calcium and cAMP (Forbes et al., 2012) (reproduced with permission from Sutherland et al. (2014)). (F) Measured chemotactic sensitivity as a function of concentration and steepness of dorsal root ganglion explants grown in gradients of nerve growth factor (Mortimer et al., 2009). (G) Chemotactic sensitivity predicted by the Bayesian model of Mortimer et al. (2009) (different y axis units from F). (H) Chemotactic sensitivity predicted by the model of Bicknell et al. (2018a), which addresses signaling pathways shared between growth and guidance (same y axis units as F).

Figure 5

Activity-Dependent Development (A) Left: Orientation selectivity map in the visual cortex of a ferret (Kaschube et al., 2010). The colors represent the orientations in the visual field to which each point in the cortex is most responsive; the white arrows in the inset highlight pinwheels, where all orientations are represented around a point. The map wavelength is about 1 mm. Right: Average pinwheel density from several animals is close to π. (B) Left: Maps formed in models with and without long-range connections: only the former produce maps resembling those seen experimentally (cf A). Right: Pinwheel density for the simulated maps is remarkably similar to that seen experimentally (Kaschube et al., 2010). (A and B) reproduced with permission from Kaschube et al. (2010). (C) Left: Simulated orientation maps with overlaid pinwheels (black dots) and ocular dominance column borders (black lines) for a normally reared animal from the model of Cloherty et al. (2016). Right: Predicted map for an animal seeing horizontal contours in one eye and vertical contours in the other during development. Reproduced from Cloherty et al. (2016). (D) Sample of oriented receptive fields produced by applying independent components analysis to natural scenes. Each gray square represents one neuron's receptive field, and the weights in that receptive field are represented by the gray scale with most weights being zero. Reproduced from Hyvärinen et al. (2009). (E) Dependence of change in synaptic strength with time Δt between pre- and post-synaptic spikes for STDP and BTDP. (F) Examples of oriented receptive fields produced by a model using an STDP learning rule. Gray scale is similar to (D). Reproduced with permission from Clopath et al. (2010). (G) (i) Retinal waves observed experimentally in ferrets at three different time points; each panel corresponds to an approximately 2 by 2-mm patch of retina. (ii) Simulated retinal waves from the model of Butts et al. (1999). (iii) Simulated retinal waves from the model of Godfrey and Swindale (2007). (iv) Simulated retinal waves from the model of Albert et al. (2008). Reproduced from Albert et al. (2008).